Intrinsic transconductance of carbon-nanotube field-effect transistors (CNTFETs) was investigated with carbon nanotubes (CNTs) grown both by laser ablation and chemical vapor deposition (CVD). The measured transconductance at a drain voltage of -1 V was 8.7 mu S for a CVD-grown CNT with a diameter of 1.5 nm. Very high intrinsic transconductance of 20 mu S was calculated by considering the contribution of parasitic resistance. Apparent and intrinsic transconductance per unit channel width are considerably larger than those for the state-of-the-art Si-MOSFETs. We expect that the performance of CNTFETs will advance further by improving CNT quality and by optimising device structures.